Apparently, the "inconsistency" noted by Gödel is simply that the
Constitution provides for its own amendment. Suber says: "He noticed
that the AC had procedural limitations but no substantive limitations;
hence it could be used to overturn the democratic institutions
described in the rest of the constitution." I am gravely
disappointed. I had been hoping for something brilliant and subtle
that only Gödel would have noticed.

M. Padgett also pointed out that the scheme I proposed for
amending the constitution, which I claimed would require only the
cooperation of a majority of both houses of Congress, 218 + 51 = 269
people in all, would actually require a filibuster-proof majority in
the Senate. He says that to be safe you would want all 100 senators
to conspire; I'm not sure why 60 would not be sufficient. (Under
current Senate rules, 60 senators can halt a filibuster.) This
would bring the total required to 218 + 60 = 278 conspirators.

He also pointed out that the complaisance of five Supreme Court
justices would give the President essentially dictatorial powers,
since any legal challenge to Presidential authority could be rejected
by the court. But this train of thought seems to have led both of us
down the same path, ending in the idea that this situation is not
really within the scope of the original question.

As a final note, I will point out what I think is a much more serious
loophole in the Constitution: if the Vice President is impeached and
tried by the Senate, then, as President of the Senate, he presides
over his own trial. Article I, section 3 contains an exception for
the trial of the President, where the Chief Justice presides instead.
But the framers inexplicably forgot to extend this exception to the
trial of the Vice President.